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Full-Text Articles in Mathematics
On Continuous Images Of Ultra-Arcs, Paul Bankston
On Continuous Images Of Ultra-Arcs, Paul Bankston
Mathematics, Statistics and Computer Science Faculty Research and Publications
Any space homeomorphic to one of the standard subcontinua of the Stone-Čech remainder of the real half-line is called an ultra-arc. Alternatively, an ultra-arc may be viewed as an ultracopower of the real unit interval via a free ultrafilter on a countable set. It is known that any continuum of weight is a continuous image of any ultra-arc; in this paper we address the problem of which continua are continuous images under special maps. Here are some of the results we present.
Ultracoproduct Continua And Their Regular Subcontinua, Paul Bankston
Ultracoproduct Continua And Their Regular Subcontinua, Paul Bankston
Mathematics, Statistics and Computer Science Faculty Research and Publications
We continue our study of ultracoproduct continua, focusing on the role played by the regular subcontinua—those subcontinua which are themselves ultracoproducts. Regular subcontinua help us in the analysis of intervals, composants, and noncut points of ultracoproduct continua. Also, by identifying two points when they are contained in the same regular subcontinua, we naturally generalize the partition of a standard subcontinuum of H⁎" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 14.4px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: …
Planarity Of Whitney Levels, Jorbe Bustamante, W. J. Charatonik, Raul Escobedo
Planarity Of Whitney Levels, Jorbe Bustamante, W. J. Charatonik, Raul Escobedo
Mathematics and Statistics Faculty Research & Creative Works
First, we characterize all locally connected continua whose all Whitney levels are planar. Second, we show by example that planarity is not a (strong) Whitney reversible property. This answers a question from Illanes-Nadler book [2].
Confluent Mappings And Arc Kelley Continua, W. J. Charatonik, Janusz R. Prajs, J. J. Charatonik
Confluent Mappings And Arc Kelley Continua, W. J. Charatonik, Janusz R. Prajs, J. J. Charatonik
Mathematics and Statistics Faculty Research & Creative Works
A Kelley continuum X, also called a continuum with the property of Kelley, such that, for each p X, each subcontinuum K containing p is approximated by arc-wise connected continua containing p, is called an arc Kelley continuum. A continuum homeomorphic to the inverse limit of locally connected continua with confluent bonding maps is said to be confluently LC-representable. The main subject of the paper is a study of deep connections between the arc Kelley continua and confluent mappings. It is shown that if a continuum X admits, for each ε > 0, a confluent ε-mapping onto a(n) (arc) Kelley continuum, …
Property Of Kelley For The Cartesian Products And Hyperspaces, W. J. Charatonik, J. J. Charatonik
Property Of Kelley For The Cartesian Products And Hyperspaces, W. J. Charatonik, J. J. Charatonik
Mathematics and Statistics Faculty Research & Creative Works
A continuum X having the property of Kelley is constructed such that neither X × [0, 1], nor the hyperspace C(X), nor small Whitney levels in C(X) have the property of Kelley. This answers several questions asked in the literature.
Chainability And Hemmingsen's Theorem, Paul Bankston
Chainability And Hemmingsen's Theorem, Paul Bankston
Mathematics, Statistics and Computer Science Faculty Research and Publications
On the surface, the definitions of chainability and Lebesgue covering dimension ⩽1 are quite similar as covering properties. Using the ultracoproduct construction for compact Hausdorff spaces, we explore the assertion that the similarity is only skin deep. In the case of dimension, there is a theorem of E. Hemmingsen that gives us a first-order lattice-theoretic characterization. We show that no such characterization is possible for chainability, by proving that if κ is any infinite cardinal and AA is a lattice base for a nondegenerate continuum, then AA is elementarily equivalent to a lattice base for a continuum Y …
The Chang-Los-Suszko Theorem In A Topological Setting, Paul Bankston
The Chang-Los-Suszko Theorem In A Topological Setting, Paul Bankston
Mathematics, Statistics and Computer Science Faculty Research and Publications
The Chang-Łoś-Suszko theorem of first-order model theory characterizes universal-existential classes of models as just those elementary classes that are closed under unions of chains. This theorem can then be used to equate two model-theoretic closure conditions for elementary classes; namely unions of chains and existential substructures. In the present paper we prove a topological analogue and indicate some applications.
Hereditarily Unicoherent Continua And Their Absolute Retracts, J. J. Charatonik, W. J. Charatonik, Janusz R. Prajs
Hereditarily Unicoherent Continua And Their Absolute Retracts, J. J. Charatonik, W. J. Charatonik, Janusz R. Prajs
Mathematics and Statistics Faculty Research & Creative Works
We investigate absolute retracts for classes of hereditarily unicoherent continua, tree-like continua, λ- dendroids, dendroids and some other related ones. The main results are: (1) the inverse limits of trees with confluent bonding mappings are absolute retracts of hereditarily unicoherent continua; (2) each tree-like continuum is embeddable in a special way in a tree-like absolute retract for the class of hereditarily unicoherent continua; (3) a dendroid is an absolute retract for hereditarily unicoherent continua if and only if it can be embedded as a retract into the Mohler-Nikiel universal smooth dendroid.
On Size Mappings, W. J. Charatonik, Alicja Samulewicz
On Size Mappings, W. J. Charatonik, Alicja Samulewicz
Mathematics and Statistics Faculty Research & Creative Works
A real-valued mapping r from the hyperspace of all compact subsets of a givenmetric space X is called a size mapping if r({x}) = 0 for x ∈ X and r(A) ≤ r(B) if a ⊂ B. We investigate what continua admit an open or a monotone size mapping. Special attention is paid to the diameter mappings.
A Degree Of Nonlocal Connectedness, J. J. Charatonik, W. J. Charatonik
A Degree Of Nonlocal Connectedness, J. J. Charatonik, W. J. Charatonik
Mathematics and Statistics Faculty Research & Creative Works
To any continuum X weassign an ordinal number (or the symbol ∞) s(X), called the degree of nonlocal connectedness of X. We show that (1) the degree cannot be increased under continuous surjections; (2) for hereditarily unicoherent continua X, the degree of a subcontinuum of X is less than or equal to s(X); (3) s(C(X)) ≤ s(X), where C(X) denotes the hyperspace of subcontinua of a continuum X. We also investigate the degrees of Cartesian products and inverse limits. As an application weconstruct an uncountable family of metric continua X homeomorphic to C(X).
Openness Of Induced Projections, J. J. Charatonik, W. J. Charatonik, Alejandro Illanes
Openness Of Induced Projections, J. J. Charatonik, W. J. Charatonik, Alejandro Illanes
Mathematics and Statistics Faculty Research & Creative Works
For continua X and Y it is shown that if the projection f : X x Y ->X has its induced mapping C(f) open, then X is C*-smooth. As a corollary, a characterization of dendrites in these terms is obtained.
Some Applications Of The Ultrapower Theorem To The Theory Of Compacta, Paul Bankston
Some Applications Of The Ultrapower Theorem To The Theory Of Compacta, Paul Bankston
Mathematics, Statistics and Computer Science Faculty Research and Publications
The ultrapower theorem of Keisler and Shelah allows such model-theoretic notions as elementary equivalence, elementary embedding and existential embedding to be couched in the language of categories (limits, morphism diagrams). This in turn allows analogs of these (and related) notions to be transported into unusual settings, chiefly those of Banach spaces and of compacta. Our interest here is the enrichment of the theory of compacta, especially the theory of continua, brought about by the importation of model-theoretic ideas and techniques.
Dendrites And Light Open Mappings, J. J. Charatonik, W. J. Charatonik, Pawel Krupski
Dendrites And Light Open Mappings, J. J. Charatonik, W. J. Charatonik, Pawel Krupski
Mathematics and Statistics Faculty Research & Creative Works
It is shown that a metric continuum X is a dendrite if and only if for every compact space Y and for every light open mapping f : Y ->f(Y ) such that X c f(Y ) there is a copy X1 of X in Y for which the restriction fjX1 : X1 ->X is a homeomorphism. Another characterization of dendrites in terms of continuous selections of multivalued functions is also obtained.
Openness And Monotoneity Of Induced Mappings, W. J. Charatonik
Openness And Monotoneity Of Induced Mappings, W. J. Charatonik
Mathematics and Statistics Faculty Research & Creative Works
It is shown that for locally connected continuum X if the induced mapping C(f) : C(X) ->C(Y) is open, then f is monotone. As a corollary it follows that if the continuum X is hereditarily locally connected and C(f) is open, then f is a homeomorphism. An example is given to show that local connectedness is essential in the result.